Determinant

Every square matrix can be associated with a real number called its determinant.

Notation

You can refer to the determinant of matrix as either det() or ||.

Determinant of a 2x2 Matrix

The determinant of the matix above is given by

Exercise

Determinant of a Triangular Matrix

The determinant of a triangular matrix is given by the product of the elements on its main diagonal.

Triangular matrix Summary

A triangular matrix can be:

  • Upper triangular: All 0's below the main diagonal.

  • Lower triangular: All 0's above the main diagonal.

  • Diagonal matrix: All 0's except for the main diagonal.

Minors and Cofactors of a matrix

The minor of a square matrix is the determinant of the matrix with row and column killed.

The cofactor is given by:

Hint

That means that the cofactors and the minors differ by just a sign(at most). The sign depends on the position of the entry.

Example

Determinant of general square Matrices

The determinant of a square matrix is the sum of the entries in the 1st row multiplied by their cofactors.

Note

I think we can do this with any row or column of the matrix.
Usually we take the row/column with the most 0's.

Example

The zero determinant

If is a square matrix, then if one of the following conditions hold:

  1. An entire row/column are equal.
  2. Two rows/columns are equal.
  3. One row/column is a multiple of another row/column
Example

How the determinant changes when using Elementary Row operations

An alternative to the cofactors method

If you don't feel like doing the thing with cofactors, you can try to turn the matrix into a triangular one, taking into consideration how the determinant changes when we apply elementary row operations.

  1. If we interchange two rows of A, the determinant chang sign.
Example

  1. If we add a multiple of a row of A to another row of A, the determinant stays the same.
Example

  1. If we multiply a row of A by a non-zero scalar, the determinant is equal to itself multiplied by a scalar.
Example


Hint

You can also use these opeations to create more zeros in a row or column, as to make the cofactor process easier.

Determinant of a Matrix product

If and are square matrices of the same size, then .

Example

Determinant of multiple of Matrix

If is a square matrix and is a scalar, then .

Example

Determinant of Invertible Matrix

A square matrix is invertible(non-singular) if and only if .

Example

Determinant of an Inverse Matrix

If is invertible, then .

Example

Determinant of Transpose

If is a square matrix, its determinant is equal to the determinant of its transpose.

Example

Properties of Determinants

If :

  1. is invertible.
  2. has only the trivial solution, 0. ( is the coefficient matrix)
  3. has unique solution whihch is . ( is the coefficient matrix)

Adjoint

If is a square matrix, the matrix of cofactors of A has the form:

The transpose of this matrix is called the adjoint of and denoted by :

Example

Inverse of a matrix using Adjoint